Confusing on GAN loss function
What is meant by both papers is that we have two agents (generator and discriminator) playing a game with the value function V defined as a sum of the expectations (i.e. an expectation of the outcome value defined as a sum of two terms, or actually a logarithm of a product...). The generator uses a strategy G encoded in the parameters of its neural network (θg), the discriminator uses a strategy D encoded in the parameters of its neural network (θd). Our goal is to (hopefully) find such a pair of strategies (a pair of parameter sets θgmin and θdmax) that produce the minimax value.

While trying to find the (θgmin, θdmax) pair using gradient descent, we actually have two loss functions: one is the loss function for G, parameterized by θg, another is the loss function for D, parameterized by θd, and we train them alternatively on minibatches together.

If you look at the Algorithm 1 in the original paper, the loss function for the discriminator is -log(D(x; θd)) - log(1 - D(G(z); θd), and the loss function for the generator is log(1 - D(G(z; θg)) (in both cases, in the original paper, x is sampled from the reference data distribution and z is sampled from noise):

The ideal value for the loss function of the discriminator is 0, otherwise it's greater than 0. The "loss" function of the generator is actually negative, but, for better gradient descent behavior, can be replaced with -log(D(G(z; θg)), which also has the ideal value for the generator at 0. It is impossible to reach zero loss for both generator and discriminator in the same GAN at the same time. However, the idea of the GAN is not to reach zero loss for any of the game agents (this is actually counterproductive), but to use that "double gradient descent" to "converge" the distribution of G(z) to the distribution of x.